Paper · May 12, 2020

Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions

Simon Brendle, Keaton Naff

math.DGarXiv:2005.05830v2
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Abstract

We extend the second part of on the uniqueness of ancient -solutions to higher dimensions. In dimensions , an ancient -solution is a nonflat, complete, ancient solution of the Ricci flow that is uniformly PIC and weakly PIC2; has bounded curvature; and is -noncollapsed. We show that the only noncompact ancient -solutions up to isometry are a family of shrinking cylinders, a quotient thereof, or the Bryant soliton.

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Submitted
May 12, 2020
Last revised
Jul 13, 2022 (v2)
Subjects
math.DG
Journal
Geom. Topol. 27 (2023) 153-226
DOI
10.2140/gt.2023.27.153

Comments: final version, to appear G&T

Ricci flowAncient solutions(Non)collapsingSphere theoremsStability & uniquenessBoundaries & non-compactRotational symmetry & warped products

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Uniqueness of convex ancient solutions to mean curvature flow in ℝ³ (S. Brendle and K. Choi, 2017)5Brendle 2017 Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions (S. Brendle and K. Choi, 2018)4Brendle 2018 Ancient solutions to the Ricci flow in dimension 3 (S. Brendle, 2018)3Brendle 2018 Ancient solutions to the Ricci flow in higher dimensions (Xiaolong Li and Yongjia Zhang, 2018)7Li 2018 Rotational symmetry of ancient solutions to the Ricci flow in dimension 3 – The compact case (S. Brendle, 2019)1Brendle 2019 Ancient solutions to the Ricci flow with isotropic curvature conditions (Jae Ho Cho and Yu Li, 2020)6Cho 2020 Uniqueness of compact ancient solutions to the higher dimensional Ricci flow (Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al., 2021)2Brendle 2021 Ancient solutions of Ricci flow with Type I curvature growth (Stephen Lynch and Andoni Royo Abrego, 2022)8Lynch 2022
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Rotational symmetry of ancient solutions to the Ricci flow in dimension – The compact case

    S. Brendle

    Topics Rotational symmetry & warped products, Ancient solutions, Boundaries & non-compact · Same author Simon Brendle · Shared terms “noncompact ancient”, “shrinking cylinder”, “Bryant soliton”

  2. Uniqueness of compact ancient solutions to the higher dimensional Ricci flow

    Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.

    Topics (Non)collapsing, Ancient solutions, Stability & uniqueness · Same authors Simon Brendle, Keaton Naff · Shared terms “higher dimensions”, “noncollapsed”

  3. Ancient solutions to the Ricci flow in dimension 3

    S. Brendle

    Topics Ancient solutions, Boundaries & non-compact · Same author Simon Brendle · Shared terms “quotient thereof”, “noncompact ancient”, “shrinking cylinder”

  4. Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions

    S. Brendle and K. ChoiWider flows

    Topics Rotational symmetry & warped products, (Non)collapsing, Ancient solutions · Same author Simon Brendle · Shared terms “higher dimensions”, “noncompact ancient”, “noncollapsed”

  5. Uniqueness of convex ancient solutions to mean curvature flow in

    S. Brendle and K. Choi

    Topics Sphere theorems, Rotational symmetry & warped products, (Non)collapsing · Same author Simon Brendle · Shared terms “noncompact ancient”, “noncollapsed”

  6. Ancient solutions to the Ricci flow with isotropic curvature conditions

    Jae Ho Cho and Yu Li

    Topics Sphere theorems, (Non)collapsing, Ancient solutions · Shared terms “weakly PIC”, “quotient thereof”, “uniformly PIC”

  7. Ancient solutions to the Ricci flow in higher dimensions

    Xiaolong Li and Yongjia Zhang

    Topics Sphere theorems, Rotational symmetry & warped products, (Non)collapsing · Shared terms “higher dimensions”, “Bryant soliton”, “noncollapsed”

  8. Ancient solutions of Ricci flow with Type I curvature growth

    Stephen Lynch and Andoni Royo Abrego

    Topics Sphere theorems, (Non)collapsing, Ancient solutions · Shared terms “PIC”, “higher dimensions”, “noncollapsed”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

  1. 12B.4
    Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery

    Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

  2. 11A.11
    Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles

    Positive Ricci curvature in dimension 3 flows to a round metric.

  3. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.